Here, \(\mathord {\in _X} \subseteq X\times \mathcal {Q} X\)  is the element-of
relation for \(X\) , and \(\overline{T}\)  is the Barr extension
of \(T\) , viz. the functor \(\overline{T}\)  on the category of sets and
relations defined on a relation \(R \subseteq X \times Y\)  by
\(\overline{T} R = \lbrace  (T\pi _1 (r), T\pi _2 (r)) \mid r\in TR\rbrace \) , where \(\pi _1: R \rightarrow X\)  and \(\pi _2: R \rightarrow Y\)  are the
projection maps (cf. [1]}). Barr [2]} (see
also Trnková [3]}) proved that \(\overline{T}\)  is a functor if and
only if \(T\)  preserves weak pullbacks.
